20777_Problem_solving_Year_6_Paterns_and_algebra_Fractions_and_decimals_Using_units_of_measurement
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YEAR 6<br />
PROBLEM-SOLVING<br />
IN MATHEMATICS<br />
Patterns <strong>and</strong> <strong>algebra</strong>/<br />
<strong>Fractions</strong> <strong>and</strong> <strong>decimals</strong>/<br />
<strong>Using</strong> <strong>units</strong> <strong>of</strong> <strong>measurement</strong><br />
Two-time winner <strong>of</strong> the Australian<br />
Primary Publisher <strong>of</strong> the <strong>Year</strong> Award
<strong>Problem</strong>-<strong>solving</strong> in mathematics<br />
(Book G)<br />
Published by R.I.C. Publications ® 2008<br />
Copyright © George Booker <strong>and</strong><br />
Denise Bond 2007<br />
RIC–<strong>20777</strong><br />
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FOREWORD<br />
Books A–G <strong>of</strong> <strong>Problem</strong>-<strong>solving</strong> in mathematics have been developed to provide a rich resource for teachers<br />
<strong>of</strong> students from the early years to the end <strong>of</strong> middle school <strong>and</strong> into secondary school. The series <strong>of</strong> problems,<br />
discussions <strong>of</strong> ways to underst<strong>and</strong> what is being asked <strong>and</strong> means <strong>of</strong> obtaining solutions have been built up to<br />
improve the problem-<strong>solving</strong> performance <strong>and</strong> persistence <strong>of</strong> all students. It is a fundamental belief <strong>of</strong> the authors<br />
that it is critical that students <strong>and</strong> teachers engage with a few complex problems over an extended period rather than<br />
spend a short time on many straightforward ‘problems’ or exercises. In particular, it is essential to allow students<br />
time to review <strong>and</strong> discuss what is required in the problem-<strong>solving</strong> process before moving to another <strong>and</strong> different<br />
problem. This book includes extensive ideas for extending problems <strong>and</strong> solution strategies to assist teachers in<br />
implementing this vital aspect <strong>of</strong> mathematics in their classrooms. Also, the problems have been constructed <strong>and</strong><br />
selected over many years’ experience with students at all levels <strong>of</strong> mathematical talent <strong>and</strong> persistence, as well as<br />
in discussions with teachers in classrooms, pr<strong>of</strong>essional learning <strong>and</strong> university settings.<br />
<strong>Problem</strong>-<strong>solving</strong> does not come easily to most people,<br />
so learners need many experiences engaging with<br />
problems if they are to develop this crucial ability. As<br />
they grapple with problem, meaning <strong>and</strong> find solutions,<br />
students will learn a great deal about mathematics<br />
<strong>and</strong> mathematical reasoning; for instance, how to<br />
organise information to uncover meanings <strong>and</strong> allow<br />
connections among the various facets <strong>of</strong> a problem<br />
to become more apparent, leading to a focus on<br />
organising what needs to be done rather than simply<br />
looking to apply one or more strategies. In turn, this<br />
extended thinking will help students make informed<br />
choices about events that impact on their lives <strong>and</strong> to<br />
interpret <strong>and</strong> respond to the decisions made by others<br />
at school, in everyday life <strong>and</strong> in further study.<br />
Student <strong>and</strong> teacher pages<br />
The student pages present problems chosen with a<br />
particular problem-<strong>solving</strong> focus <strong>and</strong> draw on a range<br />
<strong>of</strong> mathematical underst<strong>and</strong>ings <strong>and</strong> processes.<br />
For each set <strong>of</strong> related problems, teacher notes <strong>and</strong><br />
discussion are provided, as well as indications <strong>of</strong><br />
how particular problems can be examined <strong>and</strong> solved.<br />
Answers to the more straightforward problems <strong>and</strong><br />
detailed solutions to the more complex problems<br />
ensure appropriate explanations, the use <strong>of</strong> the<br />
pages, foster discussion among students <strong>and</strong> suggest<br />
ways in which problems can be extended. Related<br />
problems occur on one or more pages that extend the<br />
problem’s ideas, the solution processes <strong>and</strong> students’<br />
underst<strong>and</strong>ing <strong>of</strong> the range <strong>of</strong> ways to come to terms<br />
with what problems are asking.<br />
At the top <strong>of</strong> each teacher page, there is a statement<br />
that highlights the particular thinking that the<br />
problems will dem<strong>and</strong>, together with an indication<br />
<strong>of</strong> the mathematics that might be needed <strong>and</strong> a list<br />
<strong>of</strong> materials that could be used in seeking a solution.<br />
A particular focus for the page or set <strong>of</strong> three pages<br />
<strong>of</strong> problems then exp<strong>and</strong>s on these aspects. Each<br />
book is organised so that when a problem requires<br />
complicated strategic thinking, two or three problems<br />
occur on one page (supported by a teacher page with<br />
detailed discussion) to encourage students to find<br />
a solution together with a range <strong>of</strong> means that can<br />
be followed. More <strong>of</strong>ten, problems are grouped as a<br />
series <strong>of</strong> three interrelated pages where the level <strong>of</strong><br />
complexity gradually increases, while the associated<br />
teacher page examines one or two <strong>of</strong> the problems in<br />
depth <strong>and</strong> highlights how the other problems might be<br />
solved in a similar manner.<br />
R.I.C. Publications ® www.ricpublications.com.au <strong>Problem</strong>-<strong>solving</strong> in mathematics<br />
iii
FOREWORD<br />
Each teacher page concludes with two further aspects<br />
critical to successful teaching <strong>of</strong> problem-<strong>solving</strong>. A<br />
section on likely difficulties points to reasoning <strong>and</strong><br />
content inadequacies that experience has shown may<br />
well impede students’ success. In this way, teachers<br />
can be on the look out for difficulties <strong>and</strong> be prepared<br />
to guide students past these potential pitfalls. The<br />
final section suggests extensions to the problems to<br />
enable teachers to provide several related experiences<br />
with problems <strong>of</strong> these kinds in order to build a rich<br />
array <strong>of</strong> experiences with particular solution methods;<br />
for example, the numbers, shapes or <strong>measurement</strong>s<br />
in the original problems might change but leave the<br />
means to a solution essentially the same, or the<br />
context may change while the numbers, shapes or<br />
<strong>measurement</strong>s remain the same. Then numbers,<br />
shapes or <strong>measurement</strong>s <strong>and</strong> the context could be<br />
changed to see how the students h<strong>and</strong>le situations<br />
that appear different but are essentially the same<br />
as those already met <strong>and</strong> solved. Other suggestions<br />
ask students to make <strong>and</strong> pose their own problems,<br />
investigate <strong>and</strong> present background to the problems<br />
or topics to the class, or consider solutions at a more<br />
general level (possibly involving verbal descriptions<br />
<strong>and</strong> eventually pictorial or symbolic arguments).<br />
In this way, not only are students’ ways <strong>of</strong> thinking<br />
extended but the problems written on one page are<br />
used to produce several more problems that utilise<br />
the same approach.<br />
Mathematics <strong>and</strong> language<br />
The difficulty <strong>of</strong> the mathematics gradually increases<br />
over the series, largely in line with what is taught<br />
at the various year levels, although problem-<strong>solving</strong><br />
both challenges at the point <strong>of</strong> the mathematics<br />
that is being learned as well as provides insights<br />
<strong>and</strong> motivation for what might be learned next. For<br />
example, the computation required gradually builds<br />
from additive thinking, using addition <strong>and</strong> subtraction<br />
separately <strong>and</strong> together, to multiplicative thinking,<br />
where multiplication <strong>and</strong> division are connected<br />
conceptions. More complex interactions <strong>of</strong> these<br />
operations build up over the series as the operations<br />
are used to both come to terms with problems’<br />
meanings <strong>and</strong> to achieve solutions. Similarly, twodimensional<br />
geometry is used at first but extended<br />
to more complex uses over the range <strong>of</strong> problems,<br />
then joined by interaction with three-dimensional<br />
ideas. Measurement, including chance <strong>and</strong> data, also<br />
extends over the series from length to perimeter, <strong>and</strong><br />
from area to surface area <strong>and</strong> volume, drawing on<br />
the relationships among these concepts to organise<br />
solutions as well as giving an underst<strong>and</strong>ing <strong>of</strong> the<br />
metric system. Time concepts range from interpreting<br />
timetables using 12-hour <strong>and</strong> 24-hour clocks while<br />
investigations related to mass rely on both the concept<br />
itself <strong>and</strong> practical <strong>measurement</strong>s.<br />
The language in which the problems are expressed is<br />
relatively straightforward, although this too increases<br />
in complexity <strong>and</strong> length <strong>of</strong> expression across the books<br />
in terms <strong>of</strong> both the context in which the problems<br />
are set <strong>and</strong> the mathematical content that is required.<br />
It will always be a challenge for some students<br />
to ‘unpack’ the meaning from a worded problem,<br />
particularly as problems’ context, information <strong>and</strong><br />
meanings exp<strong>and</strong>. This ability is fundamental to the<br />
nature <strong>of</strong> mathematical problem-<strong>solving</strong> <strong>and</strong> needs to<br />
be built up with time <strong>and</strong> experiences rather than be<br />
iv<br />
<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®
FOREWORD<br />
diminished or left out <strong>of</strong> the problems’ situations. One<br />
reason for the suggestion that students work in groups<br />
is to allow them to share <strong>and</strong> assist each other with<br />
the tasks <strong>of</strong> discerning meanings <strong>and</strong> ways to tackle<br />
the ideas in complex problems through discussion,<br />
rather than simply leaping into the first ideas that<br />
come to mind (leaving the full extent <strong>of</strong> the problem<br />
unrealised).<br />
An approach to <strong>solving</strong> problems<br />
Try<br />
an approach<br />
Explore<br />
means to a solution<br />
Analyse<br />
the problem<br />
The careful, gradual development <strong>of</strong> an ability to<br />
analyse problems for meaning, organising information<br />
to make it meaningful <strong>and</strong> to make the connections<br />
among them more meaningful in order to suggest<br />
a way forward to a solution is fundamental to the<br />
approach taken with this series, from the first book<br />
to the last. At first, materials are used explicitly to<br />
aid these meanings <strong>and</strong> connections; however, in<br />
time they give way to diagrams, tables <strong>and</strong> symbols<br />
as underst<strong>and</strong>ing <strong>and</strong> experience <strong>of</strong> <strong>solving</strong> complex,<br />
engaging problems increases. As the problem forms<br />
exp<strong>and</strong>, the range <strong>of</strong> methods to solve problems<br />
is carefully extended, not only to allow students to<br />
successfully solve the many types <strong>of</strong> problems, but<br />
also to give them a repertoire <strong>of</strong> solution processes<br />
that they can consider <strong>and</strong> draw on when new<br />
situations are encountered. In turn, this allows them<br />
to explore one or other <strong>of</strong> these approaches to see<br />
whether each might furnish a likely result. In this way,<br />
when they try a particular method to solve a new<br />
problem, experience <strong>and</strong> analysis <strong>of</strong> the particular<br />
situation assists them to develop a full solution.<br />
Not only is this model for the problem-<strong>solving</strong> process<br />
helpful in <strong>solving</strong> problems, it also provides a basis for<br />
students to discuss their progress <strong>and</strong> solutions <strong>and</strong><br />
determine whether or not they have fully answered<br />
a question. At the same time, it guides teacher<br />
questions <strong>of</strong> students <strong>and</strong> provides a means <strong>of</strong> seeing<br />
underlying mathematical difficulties <strong>and</strong> ways in<br />
which problems can be adapted to suit particular<br />
needs <strong>and</strong> extensions. Above all, it provides a common<br />
framework for discussions between a teacher <strong>and</strong><br />
group or whole class to focus on the problem-<strong>solving</strong><br />
process rather than simply on the solution <strong>of</strong> particular<br />
problems. Indeed, as Alan Schoenfeld, in Steen L (Ed)<br />
Mathematics <strong>and</strong> democracy (2001), states so well, in<br />
problem-<strong>solving</strong>:<br />
getting the answer is only the beginning rather than<br />
the end … an ability to communicate thinking is<br />
equally important.<br />
We wish all teachers <strong>and</strong> students who use these<br />
books success in fostering engagement with problem<strong>solving</strong><br />
<strong>and</strong> building a greater capacity to come to<br />
terms with <strong>and</strong> solve mathematical problems at all<br />
levels.<br />
George Booker <strong>and</strong> Denise Bond<br />
R.I.C. Publications ® www.ricpublications.com.au <strong>Problem</strong>-<strong>solving</strong> in mathematics<br />
v
CONTENTS<br />
Foreword .................................................................. iii – v<br />
Contents .......................................................................... vi<br />
Introduction ........................................................... vii – xix<br />
A note on calculator use ................................................ xx<br />
Teacher notes................................................................... 2<br />
Surface area...................................................................... 3<br />
Volume <strong>and</strong> surface area................................................. 4<br />
Surface area <strong>and</strong> volume................................................. 5<br />
Teacher notes................................................................... 6<br />
The farmers market.......................................................... 7<br />
Teacher notes................................................................... 8<br />
Pr<strong>of</strong>it <strong>and</strong> loss.................................................................. 9<br />
Calculator patterns........................................................ 10<br />
Puzzle scrolls.................................................................. 11<br />
Teacher notes................................................................. 12<br />
Weather or not............................................................... 13<br />
Showtime....................................................................... 14<br />
Probably true.................................................................. 15<br />
Teacher notes................................................................. 16<br />
Wilderness explorer....................................................... 17<br />
Teacher notes................................................................. 18<br />
Changing lockers............................................................ 19<br />
Cycle days...................................................................... 20<br />
Puzzle scrolls.................................................................. 21<br />
Teacher notes................................................................. 22<br />
Office hours.................................................................... 23<br />
At the <strong>of</strong>fice................................................................... 24<br />
Out <strong>of</strong> <strong>of</strong>fice................................................................... 25<br />
Teacher notes................................................................. 26<br />
Number patterns............................................................ 27<br />
Teacher notes................................................................. 28<br />
Magic squares............................................................... 29<br />
Sudoku........................................................................... 30<br />
Alphametic puzzles........................................................ 31<br />
Teacher notes................................................................. 32<br />
At the shops................................................................... 33<br />
The plant nursery........................................................... 34<br />
On the farm.................................................................... 35<br />
Teacher notes................................................................. 36<br />
Fisherman’s wharf.......................................................... 37<br />
Teacher notes................................................................. 38<br />
Making designs.............................................................. 39<br />
Squares <strong>and</strong> rectangles................................................. 40<br />
Designer squares........................................................... 41<br />
Teacher notes................................................................. 42<br />
How many?..................................................................... 43<br />
How far?......................................................................... 44<br />
How much?..................................................................... 45<br />
Teacher notes................................................................. 46<br />
Prospect Plains............................................................... 47<br />
Teacher notes................................................................. 48<br />
Money matters............................................................... 49<br />
Scoring points................................................................ 50<br />
Puzzle scrolls.................................................................. 51<br />
Teacher notes.................................................................. 52<br />
Riding to work................................................................. 53<br />
Bike tracks....................................................................... 54<br />
Puzzle scrolls................................................................... 55<br />
Teacher notes.................................................................. 56<br />
Farm work....................................................................... 57<br />
Farm produce.................................................................. 58<br />
Selling fish...................................................................... 59<br />
Teacher notes.................................................................. 60<br />
Rolling along................................................................... 61<br />
Solutions .................................................................62–71<br />
Isometric resource page .............................................. 72<br />
0–99 board resource page ........................................... 73<br />
4-digit number exp<strong>and</strong>er resource page (x 3) .............. 74<br />
10 mm x 10 mm grid resource page ........................... 75<br />
15 mm x 15 mm grid resource page ............................ 76<br />
Triangular grid resource page ...................................... 77<br />
vi<br />
<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®
TEACHER NOTES<br />
<strong>Problem</strong>-<strong>solving</strong><br />
To use strategic thinking to solve problems<br />
Materials<br />
grid paper, counters in several different colours<br />
Focus<br />
These pages explore more complex problems in which the<br />
most difficult step is to find a way <strong>of</strong> coming to terms with<br />
what the problem is asking. <strong>Using</strong> a table or diagram to<br />
explore the situation will assist in seeing all the conditions<br />
that need to be considered.<br />
Discussion<br />
Page 19<br />
The first problem can be solved using counters on a<br />
grid or colouring the squares to see what is happening.<br />
1 2 3 4 5 6 7 8 9<br />
The number <strong>of</strong> possibilities can then be seen directly or<br />
patterns can be sought.<br />
Analysis <strong>of</strong> the patterns shows that the squares represent<br />
the factors <strong>of</strong> the number <strong>of</strong> the column in which they<br />
occur. Determining the factors <strong>of</strong> each number 1–50 shows<br />
that 48 has the most factors or entries in a column. The<br />
other questions are solved by considering prime numbers,<br />
squares <strong>of</strong> prime numbers <strong>and</strong> systematically examining<br />
the pairs <strong>of</strong> factors in each number. A discussion <strong>of</strong> prime<br />
numbers is on page 64.<br />
Page 20<br />
For the first problem, consider the distances covered each<br />
15 minutes (a table or list would help). After 15 minutes,<br />
Jane would walk 1 km <strong>and</strong> have 15 km left, while Jenny<br />
would ride 3 km <strong>and</strong> have 13 km left. After 30 minutes,<br />
Jane would walk 2 km <strong>and</strong> have 14 km left, while Jenny<br />
would ride 6 km <strong>and</strong> have 10 km left. After 45 minutes,<br />
Jane would walk 3 km <strong>and</strong> have 13 km left, while Jenny<br />
would ride 9 km <strong>and</strong> have only 7 km left. Jenny would<br />
then be home first. Jenny should leave the bicycle after 30<br />
minutes <strong>and</strong> they would both arrive home together after<br />
150 minutes. (Tables to illustrate this solution are on page<br />
64).<br />
For the second problem, organising the information on to a<br />
table will supply a solution. The slower speed requires an<br />
additional 10 km, the faster speed covers 15 km too much:<br />
hours<br />
distance @<br />
10 km/h<br />
distance<br />
to park<br />
distance @<br />
15 km/h<br />
distance<br />
to park<br />
1 10 20 15 0<br />
2 20 30 30 15<br />
3 30 40 45 30<br />
4 40 50 60 45<br />
5 50 60 75 60<br />
After 5 hours, the distance is 60 km so 12 km/h would<br />
arrive at the exact time. The third problem is done the<br />
same way – the required speed is 9.6 km/h for 5 hours.<br />
Page 21<br />
The puzzle scrolls contain a number <strong>of</strong> different problems<br />
all involving strategic thinking to find possible solutions. In<br />
most cases students will find tables, lists <strong>and</strong> diagrams are<br />
needed to manage the data while exploring the different<br />
possibilities. For example, the investigation about bread<br />
rolls could involve students list the multiples <strong>of</strong> 60 in a<br />
table with the total to make $11 <strong>and</strong> seeing which is a<br />
multiple <strong>of</strong> 85.<br />
Multiple<br />
<strong>of</strong> $0.60<br />
Possible difficulties<br />
• Not using a diagram or table to come to terms with<br />
the problem conditions<br />
• Unable to see how to connect the time cycled to the<br />
distance travelled<br />
• Considering only some aspects <strong>of</strong> the puzzle scrolls<br />
Extension<br />
• Change the numbers <strong>and</strong> the scenarios to write other<br />
problems based on the puzzle scrolls.<br />
• Use different speeds <strong>and</strong> times for the problems<br />
on p. 20.<br />
Look for a<br />
multiple <strong>of</strong> $0.85<br />
Total<br />
$.060 $10.40 $11.00<br />
$1.20 $9.80 $11.00<br />
18<br />
<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®
CHANGING LOCKERS<br />
The local college has exactly 1000 students, each <strong>of</strong> whom has a locker. The lockers<br />
are along the passageways <strong>and</strong> numbered 1–1000. To raise money for local charities,<br />
the students organised a competition with an entry fee <strong>of</strong> $1.00.<br />
All 1000 students had to run past, opening or shutting locker doors:<br />
• the first student opened the door <strong>of</strong> every locker<br />
• the second student closed every locker door with an even number<br />
• the third student changed every third locker, closing those that were open <strong>and</strong><br />
opening those that were closed<br />
• the fourth student changed every fourth locker, <strong>and</strong> so on.<br />
1. The student or students who could predict ahead <strong>of</strong> time which lockers would be open<br />
would nominate the charity that would receive the $1000.00.<br />
(a) What would you predict?<br />
(e) Use your pattern to work out which<br />
lockers would be open after all<br />
1000 students had run past.<br />
(b) Would any lockers remain open<br />
after 10 students had passed along<br />
the rows <strong>of</strong> lockers?<br />
(f) Can you find some lockers that<br />
only changed twice?<br />
(c) Which lockers would be open<br />
after 50 students had changed 50<br />
lockers?<br />
(g) Can you see a pattern for the<br />
numbers on these lockers?<br />
(d) Can you see a pattern for the<br />
numbers <strong>of</strong> the open lockers?<br />
(h) What is the largest number on a<br />
locker that changed only twice?<br />
R.I.C. Publications ® www.ricpublications.com.au <strong>Problem</strong>-<strong>solving</strong> in mathematics<br />
19
CYCLE DAYS<br />
Jane <strong>and</strong> Jenny rode their bicycles to the bay for a picnic. On the way back, when they<br />
were 16 km from home, the front wheel on Jane’s bicycle was damaged in a large<br />
pothole <strong>and</strong> could no longer be ridden. In order for them both to get home in good<br />
time, they decided to share the walking <strong>and</strong> bike riding.<br />
1. At first, Jane would walk while Jenny would ride her bicycle. After some time, Jenny<br />
would leave her bicycle at the side <strong>of</strong> the road <strong>and</strong> continue on foot. When Jane<br />
reached the bicycle, she would then ride it home. Jane walks at 4 km per hour <strong>and</strong><br />
cycles at 10 km per hour, while Jenny walks at 5 km per hour <strong>and</strong> cycles at 12 km per<br />
hour. How long should Jenny ride her bicycle for both <strong>of</strong> them to arrive home at the<br />
same time?<br />
2. When Jane’s bicycle was repaired, they decided to go for another bike ride. This time,<br />
they would set <strong>of</strong>f separately <strong>and</strong> meet at the Jetty at midday, then set up their picnic<br />
in the park next door. Jenny knew that if she took it easy <strong>and</strong> rode at 10 km per hour,<br />
she would be an hour late, but if she really pushed herself <strong>and</strong> rode at 15 km per hour,<br />
she would arrive an hour early. At what speed should she ride in order to arrive just on<br />
time?<br />
3. Jane did not have quite as far to travel so she knew she did not have to ride as hard as<br />
Jenny. If she rode at 12 km per hour, she would arrive an hour early, but if she rode at<br />
8 km per hour, she would be an hour late. At what speed should Jane ride to get there<br />
at the same time as Jenny?<br />
4. At what time did each girl leave her home?<br />
20<br />
<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®
PUZZLE SCROLLS<br />
1. $65 000 is divided among 5<br />
brothers with each one getting<br />
$3000 more than their younger<br />
sibling. How much will the<br />
youngest brother get?<br />
2. A rectangular table is twice as<br />
long as it is wide. If it was 3 m<br />
shorter <strong>and</strong> 3 m wider it would<br />
be a square. What size is the<br />
table?<br />
3. How many blocks would you<br />
need to make this stack <strong>of</strong> cubes<br />
into a cube 6 blocks high?<br />
4. I spent $11.00 at the bakery<br />
buying bread rolls for 60c <strong>and</strong><br />
85c. How many <strong>of</strong> each roll did I<br />
buy?<br />
5. How <strong>of</strong>ten in a 12-hour period<br />
does the sum <strong>of</strong> the digits on a<br />
digital clock equal 9?<br />
6. A triangle has a perimeter <strong>of</strong> 80<br />
cm. Two <strong>of</strong> its sides are equal<br />
<strong>and</strong> the third side is 8 cm more<br />
than the equal sides.<br />
03:24<br />
ON<br />
OFF<br />
GIGA-BLASTER<br />
What is the length <strong>of</strong> the third<br />
side?<br />
R.I.C. Publications ® www.ricpublications.com.au <strong>Problem</strong>-<strong>solving</strong> in mathematics<br />
21
SOLUTIONS<br />
Note: Many solutions are written statements rather than just numbers. This is to encourage teachers <strong>and</strong> students to solve<br />
problems in this way.<br />
CHANGING LOCKERS................................................ page 19<br />
1. (a) Answers will vary.<br />
Colour squares or place counters on a grid to show<br />
when a door is open:<br />
A door is changed an odd number <strong>of</strong> times to<br />
remain open. These locker numbers are square<br />
number (one number as a factor twice, so an odd<br />
number <strong>of</strong> factors): 1, 4, 9, 16, 25, 49, ...<br />
A prime number has only two factors <strong>and</strong> these<br />
lockers will always be closed.<br />
(b) 1, 4, 9<br />
(c) 1, 4, 9, 16, 25, 49<br />
(d) Square numbers have an odd number <strong>of</strong> factors <strong>and</strong><br />
these lockers will be open.<br />
(e) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169,<br />
225, 256, 289, 324, 361, 400, 441, 484, 529, 576,<br />
625, 676, 729, 784, 841, 900, 961<br />
(f) 2, 3, 5, 7, 11, 13<br />
(g)<br />
(h)<br />
1 2 3 4 5 6 7 8 9<br />
They are prime numbers <strong>and</strong> have only 2 factors<br />
The Sieve <strong>of</strong> Eratosthenes shows that after 2 <strong>and</strong><br />
3, all primes must be 1 more or less than a multiple<br />
<strong>of</strong> 6 (although some <strong>of</strong> these numbers are not<br />
prime – this was discussed in Book F).<br />
1 2 3 4 5 6 7<br />
8 9 10 11 12 13<br />
14 15 16 17 18 19<br />
20 21 22 23 24 25<br />
26 27 28 29 30 31<br />
32 33 34<br />
=166 x 6 = 996 is the closest multiple <strong>of</strong> 6 to 1000,<br />
so 997 is the largest prime.<br />
997 is the largest locker number that is only<br />
changed twice.<br />
CYCLE DAYS............................................................... page 20<br />
1. Looking at their rates <strong>of</strong> walking <strong>and</strong> cycling, in 15<br />
minutes Jane walks 1 km <strong>and</strong> Jenny cycles 3 km:<br />
Jane walking<br />
minutes<br />
Jenny cycling<br />
distance<br />
walked<br />
distance left to<br />
cycle<br />
They can arrive home together after 2 hours 30 minutes.<br />
Jenny cycles for 30 minutes <strong>and</strong> walks the remaining 10<br />
km in 2 hours. Jane walks for 90 minutes to reach the<br />
bicycle <strong>and</strong> cycles the remaining 10 km in 1 hour.<br />
Jenny should ride her bicycle for 30 minutes<br />
time taken<br />
15 1 15 1 hour 45 minutes<br />
30 2 14 1 hour 54 minutes<br />
45 3 13 2 hours 3 minutes<br />
60 4 12 2 hours 12 minutes<br />
75 5 11 2 hours 21 minutes<br />
90 6 10 2 hours 30 minutes<br />
minutes<br />
distance<br />
cycled<br />
distance left to<br />
walk<br />
time taken<br />
15 3 13 2 hours 51 minutes<br />
30 6 10 2 hours 30 minutes<br />
45 9 7 2 hours 90 minutes<br />
60 12 4 1 hour 48 minutes<br />
Jane 90 minutes walking 1 hour cycling<br />
km 0 1 2 3 4 5 6 7 8 9 10 11 12 13 15 15 16<br />
Jenny 30 minutes cycling 2 hours walking<br />
2. The slower speed requires an additional 10 km, the<br />
faster speed covers 15 km too much:<br />
hours<br />
distance @<br />
10km/her<br />
distance to<br />
park<br />
distance @<br />
15 km/her<br />
distance<br />
to park<br />
1 10 20 15 0<br />
2 20 30 30 15<br />
3 30 40 45 30<br />
4 40 50 60 45<br />
5 50 60 75 60<br />
After 5 hours, the distance to the park at each speed is<br />
the same – 60 km.<br />
She would arrive at the exact time cycling at<br />
12 km per hour.<br />
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<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®
SOLUTIONS<br />
Note: Many solutions are written statements rather than just numbers. This is to encourage teachers <strong>and</strong> students to solve<br />
problems in this way.<br />
3. The slower speed requires an additional 8 km, the faster<br />
speed cover 12 km too much:<br />
hours<br />
distance @<br />
10km/her<br />
distance to<br />
park<br />
distance @<br />
15 km/her<br />
distance<br />
to park<br />
1 10 20 15 0<br />
2 20 30 30 15<br />
3 30 40 45 30<br />
4 40 50 60 45<br />
5 50 60 75 60<br />
After 5 hours, the distance to the park at each speed is<br />
the same – 48 km.<br />
She would arrive at the exact time cycling at 9.6 km per<br />
hour<br />
4. Both Jenny <strong>and</strong> Jane should leave at 7:00 am<br />
PUZZLE SCROLLS ...................................................... page 21<br />
1. $7000<br />
2. width 6 m, length 12 m<br />
3. 210 blocks<br />
4. 7 @ 60 c, 8 @ 85 c<br />
5. 57 times. Student answers will vary <strong>and</strong> it is unlikely<br />
they will find them all.<br />
6. 32 cm<br />
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65
10 mm x 10 mm GRID RESOURCE PAGE<br />
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75
15 mm x 15 mm GRID RESOURCE PAGE<br />
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<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®